:: PDIFF_1 semantic presentation

begin

definition
let i, n be ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ;
func proj (i,n) -> ( ( Function-like quasi_total ) ( non empty Relation-like REAL n : ( ( ) ( ) Element of i : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL n : ( ( ) ( ) Element of i : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) means :: PDIFF_1:def 1
 for x being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(n : ( ( ) ( ) Element of i : ( ( ) ( ) NORMSTR ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( ) ( ) Element of i : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) holds it : ( ( Function-like quasi_total ) ( Relation-like K7(i : ( ( ) ( ) NORMSTR ) ,i : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) -defined i : ( ( ) ( ) NORMSTR ) -valued Function-like quasi_total ) Element of K6(K7(K7(i : ( ( ) ( ) NORMSTR ) ,i : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ,i : ( ( ) ( ) NORMSTR ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) . i : ( ( ) ( ) NORMSTR ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ;
end;

theorem :: PDIFF_1:1
( ( for i, n being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) st i : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) in Seg n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V165() V166() V167() V168() V169() V170() ) Element of K6(NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) holds
( dom (proj (i : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) Element of K6((REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( ) set ) ) = REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) & rng (proj (i : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ,n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL b2 : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( V165() V166() V167() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) = REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) & ( for x being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) holds
( (proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . <*x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) & ((proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ") : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) . x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) = <*x : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ) ) ;

theorem :: PDIFF_1:2
( (proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) " : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) is ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) & (proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) " : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) is one-to-one & dom ((proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ") : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) = REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) & rng ((proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ") : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) : ( ( ) ( ) set ) = REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) & ex g being ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st
( g : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) is bijective & (proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) " : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) = g : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ) ) ;

registration
cluster proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -> Function-like quasi_total bijective ;
end;

definition
let g be ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
func <>* g -> ( ( Function-like ) ( Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) equals :: PDIFF_1:def 2
(((proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ") : ( ( Relation-like Function-like ) ( Relation-like Function-like one-to-one ) set ) * g : ( ( ) ( ) set ) ) : ( ( Relation-like ) ( Relation-like ) set ) * (proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( Relation-like ) ( Relation-like ) set ) ;
end;

definition
let n be ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ;
let g be ( ( Function-like ) ( Relation-like REAL n : ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
func <>* g -> ( ( Function-like ) ( Relation-like REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) equals :: PDIFF_1:def 3
((proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ") : ( ( Relation-like Function-like ) ( Relation-like Function-like one-to-one ) set ) * g : ( ( ) ( ) set ) : ( ( Relation-like ) ( Relation-like ) set ) ;
end;

definition
let i, n be ( ( natural ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative ) Nat) ;
func Proj (i,n) -> ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V2() ) set ) ) means :: PDIFF_1:def 4
 for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) holds it : ( ( Function-like quasi_total ) ( Relation-like K7(i : ( ( ) ( ) set ) ,i : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined i : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(i : ( ( ) ( ) set ) ,i : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,i : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) = <*((proj (i : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . x : ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ;
end;

definition
let i be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let x be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ;
func reproj (i,x) -> ( ( Relation-like Function-like ) ( Relation-like Function-like ) Function) means :: PDIFF_1:def 5
( dom it : ( ( Function-like quasi_total ) ( Relation-like K7(i : ( ( ) ( ) set ) ,i : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined i : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(i : ( ( ) ( ) set ) ,i : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,i : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) set ) = REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) & ( for r being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) holds it : ( ( Function-like quasi_total ) ( Relation-like K7(i : ( ( ) ( ) set ) ,i : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined i : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(i : ( ( ) ( ) set ) ,i : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,i : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . r : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) = Replace (x : ( ( ) ( ) set ) ,i : ( ( ) ( ) set ) ,r : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) );
end;

definition
let n, i be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let x be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;
:: original: reproj
redefine func reproj (i,x) -> ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;
end;

definition
let n, i be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let x be ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) ;
func reproj (i,x) -> ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty ) set ) ) means :: PDIFF_1:def 6
 for r being ( ( ) ( ) Element of ( ( ) ( non empty V2() ) set ) ) ex q being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ex y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(n : ( ( ) ( ) set ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st
( r : ( ( ) ( ) Element of ( ( ) ( non empty V2() ) set ) ) = <*q : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) & y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) = x : ( ( Function-like quasi_total ) ( Relation-like K7(n : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(n : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & it : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) . r : ( ( ) ( ) Element of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) ) = (reproj (i : ( ( ) ( ) set ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) . q : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) Element of REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) );
end;

definition
let m, n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) ;
let x be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;
pred f is_differentiable_in x means :: PDIFF_1:def 7
 ex g being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ex y being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st
( f : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) & x : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = y : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) & g : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in y : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) );
end;

definition
let m, n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) ;
let x be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;
assume f : ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) is_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;
func diff (f,x) -> ( ( Function-like quasi_total ) ( non empty Relation-like REAL m : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL m : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) means :: PDIFF_1:def 8
 ex g being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ex y being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st
( f : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) & x : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = y : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) & it : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = diff (g : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ,y : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) ,(REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) );
end;

theorem :: PDIFF_1:3
for I being ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) = (proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) " : ( ( Relation-like Function-like ) ( Relation-like Function-like one-to-one ) set ) holds
( ( for x being ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) )
for y being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) st x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) = I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) . y : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) Element of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) holds
||.x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = abs y : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) & ( for x, y being ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) )
for a, b being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) st x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) = I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) . a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) Element of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) & y : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) = I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) . b : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) Element of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) holds
x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) + y : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) = I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) . (a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) + b : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) Element of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ) & ( for x being ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) )
for y being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) )
for a being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) = I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) . y : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) Element of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) holds
a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) * x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) = I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) . (a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) * y : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) Element of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ) & ( for x being ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) )
for a being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) st x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) = I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) . a : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) Element of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) holds
- x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) = I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) . (- a : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) Element of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ) & ( for x, y being ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) )
for a, b being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) st x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) = I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) . a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) Element of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) & y : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) = I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) . b : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) Element of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) holds
x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) - y : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) = I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) . (a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) - b : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) Element of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ) ) ;

theorem :: PDIFF_1:4
for J being ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) st J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) holds
( ( for x being ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) )
for y being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) st J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = y : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) holds
||.x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) .|| : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = abs y : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) & ( for x, y being ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) )
for a, b being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) st J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) & J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . y : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = b : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) holds
J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . (x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) + y : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) + b : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) & ( for x being ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) )
for y being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) )
for a being ( ( ) ( V11() real ext-real ) Real) st J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = y : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) holds
J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . (a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) * x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) * y : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) & ( for x being ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) )
for a being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) st J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = a : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) holds
J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . (- x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = - a : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) & ( for x, y being ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) )
for a, b being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) st J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) & J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . y : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = b : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) holds
J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . (x : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) - y : ( ( ) ( ) VECTOR of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = a : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) - b : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ) ;

theorem :: PDIFF_1:5
for I being ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) st I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) = (proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) " : ( ( Relation-like Function-like ) ( Relation-like Function-like one-to-one ) set ) & J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) holds
( ( for R being ( ( Function-like RestFunc-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) holds (J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) * R : ( ( Function-like quasi_total V158( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) V159( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total V158( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) V159( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) ) LinearOperator of REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) * I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) ) & ( for L being ( ( Function-like quasi_total V158( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) V159( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total V158( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) V159( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) ) LinearOperator of REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) holds (J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) * L : ( ( Function-like quasi_total V158( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) V159( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total V158( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) V159( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) ) LinearOperator of REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) * I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is ( ( Function-like linear ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) ) ) ;

theorem :: PDIFF_1:6
for I being ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) )
for J being ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) st I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) = (proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) " : ( ( Relation-like Function-like ) ( Relation-like Function-like one-to-one ) set ) & J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) holds
( ( for R being ( ( Function-like RestFunc-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued RestFunc-like ) RestFunc) holds (I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) * R : ( ( Function-like linear ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( Function-like ) ( Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is ( ( Function-like RestFunc-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like RestFunc-like ) RestFunc of REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) ) & ( for L being ( ( Function-like linear ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) holds (I : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) * L : ( ( Function-like linear ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued linear ) LinearFunc) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) Element of K6(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * J : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( Function-like ) ( Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ,(REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is ( ( Function-like quasi_total V158( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) V159( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) Lipschitzian ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total V158( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) V159( REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) Lipschitzian ) LinearOperator of REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) , REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ) ) ) ;

theorem :: PDIFF_1:7
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for y being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) st f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = <>* g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) & x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = <*y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) holds
( g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_differentiable_in y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) & diff (g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) = (((proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) * (diff (f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ,(REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) : ( ( Relation-like ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) * ((proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ") : ( ( Relation-like Function-like ) ( Relation-like Function-like one-to-one ) set ) ) : ( ( Relation-like ) ( Relation-like Function-like complex-valued ext-real-valued real-valued ) set ) . 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;

theorem :: PDIFF_1:8
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for y being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) st f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = <>* g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) & x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = <*y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) & g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_differentiable_in y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) & (diff (f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ,(REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) . <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued natural-valued ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = <*(diff (g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;

theorem :: PDIFF_1:9
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for y being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) st f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = <>* g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) & x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = <*y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) iff g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_differentiable_in y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;

theorem :: PDIFF_1:10
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for y being ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) st f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = <>* g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) & x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = <*y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) & f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) holds
(diff (f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ,(REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) . <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued natural-valued ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = <*(diff (g : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ;

begin

definition
let n, m be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let i be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ;
let x be ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ;
pred f is_partial_differentiable_in x,i means :: PDIFF_1:def 9
f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * (reproj (i : ( ( Function-like quasi_total ) ( Relation-like K7(n : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(n : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( Function-like quasi_total ) ( Relation-like n : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(n : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in (Proj (i : ( ( Function-like quasi_total ) ( Relation-like K7(n : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(n : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,m : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V2() ) set ) ) . x : ( ( Function-like quasi_total ) ( Relation-like n : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(n : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) ;
end;

definition
let m, n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let i be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ;
let x be ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ;
func partdiff (f,x,i) -> ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) equals :: PDIFF_1:def 10
 diff ((f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * (reproj (i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,((Proj (i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,m : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V2() ) set ) ) . x : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ,(REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let i be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let x be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;
pred f is_partial_differentiable_in x,i means :: PDIFF_1:def 11
f : ( ( Function-like quasi_total ) ( Relation-like K7(n : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(n : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * (reproj (i : ( ( ) ( ) set ) ,x : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in (proj (i : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . x : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ;
end;

definition
let n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let i be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let x be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;
func partdiff (f,x,i) -> ( ( real ) ( V11() real ext-real ) number ) equals :: PDIFF_1:def 12
 diff ((f : ( ( Function-like quasi_total ) ( Relation-like K7(n : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(n : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * (reproj (i : ( ( ) ( ) set ) ,x : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,((proj (i : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . x : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined n : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,n : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ;
end;

definition
let n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let i be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ;
let x be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;
:: original: partdiff
redefine func partdiff (f,x,i) -> ( ( ) ( V11() real ext-real ) Real) ;
end;

theorem :: PDIFF_1:11
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds Proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty V2() ) set ) ) = ((proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ") : ( ( Relation-like Function-like ) ( Relation-like Function-like one-to-one ) set ) * (proj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( Relation-like ) ( Relation-like Function-like ) set ) ;

theorem :: PDIFF_1:12
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) holds
(reproj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) * (proj (1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like one-to-one total quasi_total onto bijective complex-valued ext-real-valued real-valued ) Function of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( Function-like ) ( non empty Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Element of K6(K7((REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ,(REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = reproj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty V2() ) set ) ) ;

theorem :: PDIFF_1:13
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = <>* g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) & x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) holds
<>* (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) * (reproj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) = f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) * (reproj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty V2() ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_1:14
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = <>* g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) & x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) iff g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: PDIFF_1:15
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = <>* g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) & x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) & f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(partdiff (f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued natural-valued ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = <*(partdiff (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ;

definition
let m, n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let i be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) ;
let x be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;
pred f is_partial_differentiable_in x,i means :: PDIFF_1:def 13
 ex g being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ex y being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st
( f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) & x : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = y : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) & g : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in y : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) );
end;

definition
let m, n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let i be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) ;
let x be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;
assume f : ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
func partdiff (f,x,i) -> ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(n : ( ( ) ( ) set ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) means :: PDIFF_1:def 14
 ex g being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) ex y being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st
( f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = g : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) & x : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) = y : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) & it : ( ( Function-like ) ( Relation-like i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like ) Element of K6(K7(i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (partdiff (g : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ,y : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued natural-valued ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) );
end;

theorem :: PDIFF_1:16
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for F being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for G being ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st F : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = G : ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) & x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) holds
( F : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) iff G : ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: PDIFF_1:17
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for F being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for G being ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st F : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = G : ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) & x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) & F : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(partdiff (F : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued natural-valued ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = partdiff (G : ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b3 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;

theorem :: PDIFF_1:18
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) )
for g1 being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) st g1 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) = <>* g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) holds
( g1 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) iff g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: PDIFF_1:19
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) )
for g1 being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) st g1 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) = <>* g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) & g1 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
partdiff (g1 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) = <*(partdiff (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Real) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued ) FinSequence of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ;

begin

definition
let m, n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let i, j be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ;
let x be ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ;
pred f is_partial_differentiable_in x,i,j means :: PDIFF_1:def 15
((Proj (j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,n : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V2() ) set ) ) * f : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * (reproj (i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( Function-like ) ( Relation-like i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like ) Element of K6(K7(i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in (Proj (i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,m : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V2() ) set ) ) . x : ( ( Function-like ) ( Relation-like i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like ) Element of K6(K7(i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) ;
end;

definition
let m, n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let i, j be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ;
let x be ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ;
func partdiff (f,x,i,j) -> ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) equals :: PDIFF_1:def 16
 diff ((((Proj (j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,n : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V2() ) set ) ) * f : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * (reproj (i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( Function-like ) ( Relation-like i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like ) Element of K6(K7(i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty ) set ) ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,((Proj (i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,m : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty ) set ) , ( ( ) ( non empty V2() ) set ) ) . x : ( ( Function-like ) ( Relation-like i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like ) Element of K6(K7(i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) Element of the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ,(REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ;
end;

definition
let m, n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let i, j be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let h be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) ;
let z be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;
pred h is_partial_differentiable_in z,i,j means :: PDIFF_1:def 17
((proj (j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,n : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) * h : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL m : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL m : ( ( ) ( ) set ) ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) * (reproj (i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,z : ( ( Function-like ) ( Relation-like i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like ) Element of K6(K7(i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL m : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL m : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in (proj (i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,m : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL m : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL m : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . z : ( ( Function-like ) ( Relation-like i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like ) Element of K6(K7(i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ;
end;

definition
let m, n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let i, j be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let h be ( ( Function-like ) ( Relation-like REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) ;
let z be ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;
func partdiff (h,z,i,j) -> ( ( real ) ( V11() real ext-real ) number ) equals :: PDIFF_1:def 18
 diff ((((proj (j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,n : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL n : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) * h : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( Relation-like REAL m : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL m : ( ( ) ( ) set ) ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) * (reproj (i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,z : ( ( Function-like ) ( Relation-like i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like ) Element of K6(K7(i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL m : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL m : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,((proj (i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,m : ( ( ) ( ) set ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL m : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL m : ( ( ) ( ) set ) : ( ( ) ( ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) . z : ( ( Function-like ) ( Relation-like i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like ) Element of K6(K7(i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,j : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ;
end;

theorem :: PDIFF_1:20
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for F being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for G being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st F : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = G : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) & x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) holds
( F : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) iff G : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) is_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ) ;

theorem :: PDIFF_1:21
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for F being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for G being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st F : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = G : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) & x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) & F : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) holds
diff (F : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ,(REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) = diff (G : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;

theorem :: PDIFF_1:22
for m, n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for j, i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for h being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for z being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = h : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) & x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = z : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) holds
((Proj (j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty V2() ) set ) ) * f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * (reproj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty V2() ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = <>* (((proj (j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Function of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) , REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) * h : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) * (reproj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,z : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like total quasi_total ) Function of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) , REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ) : ( ( Function-like ) ( Relation-like REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) ;

theorem :: PDIFF_1:23
for n, m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for h being ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for z being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = h : ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) & x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = z : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) holds
( f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) iff h : ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in z : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: PDIFF_1:24
for n, m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i, j being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for h being ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) )
for z being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) = h : ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) & x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) = z : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) & f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
(partdiff (f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) . <*1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) *> : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like FinSequence-like complex-valued ext-real-valued real-valued natural-valued ) FinSequence of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) = <*(partdiff (h : ( ( Function-like ) ( Relation-like REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -valued Function-like ) PartFunc of ,) ,z : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,j : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( real ) ( V11() real ext-real ) number ) *> : ( ( Relation-like Function-like ) ( Relation-like Function-like ) set ) ;

definition
let m, n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let i be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
pred f is_partial_differentiable_on X,i means :: PDIFF_1:def 19
( X : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) c= dom f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( Relation-like ) Element of K6( the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) holds
f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) | X : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) , the carrier of (REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) );
end;

theorem :: PDIFF_1:25
for n, m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for X being ( ( ) ( ) set )
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) st f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( ) set ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
X : ( ( ) ( ) set ) is ( ( ) ( ) Subset of ) ;

definition
let m, n be ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let i be ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
let f be ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ;
let X be ( ( ) ( ) set ) ;
assume f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_on X : ( ( ) ( ) set ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ;
func f `partial| (X,i) -> ( ( Function-like ) ( Relation-like the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) -defined the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ,(REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) -valued Function-like ) PartFunc of ,) means :: PDIFF_1:def 20
( dom it : ( ( Function-like ) ( Relation-like i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like ) Element of K6(K7(i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) Element of K6( the carrier of (REAL-NS m : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) : ( ( ) ( non empty ) set ) ) : ( ( ) ( ) set ) ) = X : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( ) Point of ( ( ) ( non empty ) set ) ) st x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) in X : ( ( Function-like quasi_total ) ( Relation-like m : ( ( ) ( ) set ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like total quasi_total complex-valued ext-real-valued real-valued ) Element of K6(K7(m : ( ( ) ( ) set ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) holds
it : ( ( Function-like ) ( Relation-like i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -defined f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) -valued Function-like ) Element of K6(K7(i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) /. x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ,(REAL-NS n : ( ( ) ( ) set ) ) : ( ( non empty strict ) ( non empty strict ) NORMSTR ) )) : ( ( non empty ) ( non empty ) NORMSTR ) : ( ( ) ( non empty ) set ) ) = partdiff (f : ( ( Function-like quasi_total ) ( Relation-like K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( Function-like quasi_total ) ( Relation-like K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) -defined m : ( ( ) ( ) set ) -valued Function-like quasi_total ) Element of K6(K7(K7(m : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ,m : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) ) );
end;

theorem :: PDIFF_1:26
for n, m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) holds
( (f1 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * (reproj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty V2() ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (f1 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) * (reproj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty V2() ) set ) ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) + (f2 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) * (reproj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty V2() ) set ) ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) & (f1 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * (reproj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty V2() ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (f1 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) * (reproj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty V2() ) set ) ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) - (f2 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) * (reproj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty V2() ) set ) ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: PDIFF_1:27
for n, m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) holds r : ( ( ) ( V11() real ext-real ) Real) (#) (f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) * (reproj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty V2() ) set ) ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) = (r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) * (reproj (i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) )) : ( ( Function-like quasi_total ) ( non empty Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like total quasi_total ) Function of ( ( ) ( non empty V2() ) set ) , ( ( ) ( non empty V2() ) set ) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: PDIFF_1:28
for n, m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) st f1 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & f2 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f1 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & partdiff ((f1 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) + f2 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) = (partdiff (f1 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) + (partdiff (f2 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ,(REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: PDIFF_1:29
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for g1, g2 being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st g1 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & g2 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( g1 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) + g2 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & partdiff ((g1 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) + g2 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Real) = (partdiff (g1 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Real) + (partdiff (g2 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;

theorem :: PDIFF_1:30
for n, m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for f1, f2 being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) st f1 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & f2 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( f1 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & partdiff ((f1 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) - f2 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) = (partdiff (f1 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) - (partdiff (f2 : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ,(REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: PDIFF_1:31
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for g1, g2 being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st g1 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & g2 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( g1 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) - g2 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & partdiff ((g1 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) - g2 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Real) = (partdiff (g1 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Real) - (partdiff (g2 : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;

theorem :: PDIFF_1:32
for n, m being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,)
for x being ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) st f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) is_partial_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_in x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & partdiff ((r : ( ( ) ( V11() real ext-real ) Real) (#) f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) Element of K6(K7( the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) , the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) * (partdiff (f : ( ( Function-like ) ( Relation-like the carrier of (REAL-NS b2 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -defined the carrier of (REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty V2() ) set ) -valued Function-like ) PartFunc of ,) ,x : ( ( ) ( ) Point of ( ( ) ( non empty V2() ) set ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( Relation-like Function-like ) Point of ( ( ) ( non empty ) set ) ) : ( ( ) ( Relation-like Function-like ) Element of the carrier of (R_NormSpace_of_BoundedLinearOperators ((REAL-NS 1 : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) ,(REAL-NS b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( non empty strict ) ( non empty V50() V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() strict RealNormSpace-like V161() ) NORMSTR ) )) : ( ( non empty ) ( non empty V71() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V106() V107() RealNormSpace-like V161() ) NORMSTR ) : ( ( ) ( non empty ) set ) ) ) ;

theorem :: PDIFF_1:33
for n being ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for i being ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) )
for r being ( ( ) ( V11() real ext-real ) Real)
for g being ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,)
for y being ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL n : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) st g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) is_partial_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) is_partial_differentiable_in y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) & partdiff ((r : ( ( ) ( V11() real ext-real ) Real) (#) g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ) : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) Element of K6(K7((REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ,REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( complex-valued ext-real-valued real-valued ) set ) ) : ( ( ) ( ) set ) ) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Real) = r : ( ( ) ( V11() real ext-real ) Real) * (partdiff (g : ( ( Function-like ) ( Relation-like REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like complex-valued ext-real-valued real-valued ) PartFunc of ,) ,y : ( ( ) ( Relation-like NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) -defined REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) -valued Function-like V40(b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) ) FinSequence-like complex-valued ext-real-valued real-valued ) Element of REAL b1 : ( ( non empty ) ( non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real positive non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty FinSequence-membered ) FinSequenceSet of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ,i : ( ( ) ( epsilon-transitive epsilon-connected ordinal natural V11() real ext-real non negative V139() V164() V165() V166() V167() V168() V169() V170() ) Element of NAT : ( ( ) ( non empty epsilon-transitive epsilon-connected ordinal V165() V166() V167() V168() V169() V170() V171() ) Element of K6(REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) : ( ( ) ( ) set ) ) ) )) : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V33() V165() V166() V167() V171() ) set ) ) ) ;